Let's see you explain it simply for your rocket.
m
r is the mass of the physical rocket, with no fuel in. v
r is the velocity of the rocket (unknown). m
f is the mass of the fuel. The fuel exhaust leaves the rocket with some velocity: we call that v
f.
Aside, for completeness' sake. Ignore if you want. Technically, it might be easier to define m
r to be the mass of the rocket and some amount of the fuel, and only be concerned with the fuel exhaust over a limited amount of time. The end result will be very much the same.
Conservation of momentum states that m
1u
1 + m
2u
2 = m
1v
1 + m
2v
2 where u is the initial velocity, v the final velocity.
We know that a rocket starts at rest, and that the fuel exhaust comes out from the rocket. We'll define the direction of the fuel to be negative: it can be positive if you want, this just makes it neater. All that matters is that the velocity is non-zero.
Now, it's just a matter of applying the formula:
0(m
r + m
f) = m
rv
r - m
fv
fClearly, the left hand side is zero, so:
m
rv
r = m
fv
fWe know that fuel has some mass, and some (high) velocity, so the right hand side is not zero: so the left hand side cannot be zero. That is, the velocity of the rocket must exist.
I mean, if you want to be really technical it's closer to
(m
r+ (1-k)m
f)v
r = km
fv
fFor some constant k between 0 and 1, determining how much of the fuel has been consumed, but still. The right hand side is still non-zero, so v
r must be as well. This holds in isolated, or approximately isolated, systems.